Weakly Symmetric Pseudo-Riemannian Nilmanifolds

TL;DR

This paper classifies weakly symmetric pseudo-Riemannian nilmanifolds G/H, extending previous classifications from semisimple to nilpotent cases, revealing many new examples and analyzing automorphism extensions.

Contribution

It extends the classification of weakly symmetric pseudo-Riemannian manifolds to nilmanifolds, providing new examples and detailed automorphism extension analysis.

Findings

Many new weakly symmetric pseudo-Riemannian nilmanifolds identified

Criteria for automorphism extension to isometries established

Classification results summarized in comprehensive tables

Abstract

In an earlier paper we developed the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetric manifolds of Lorentz signature $(n-1,1)$ and trans--lorentzian signature $(n-2,2)$. Here we work out the classification of weakly symmetric pseudo--riemannian nilmanifolds $G/H$ from the classification for the case $G = N\rtimes H$ with $H$ compact and $N$ nilpotent. It turns out that there is a plethora of new examples that merit further study. Starting with that riemannian case, we see just when a given involutive automorphism of $H$ extends to an involutive automorphism of $G$, and we show that any two such extensions result in isometric pseudo--riemannian nilmanifolds. The results are tabulated in the last two sections of the paper.